Well, I recently bought Football Manager 2010, and tried to install it on this computer.
Every time I try, I get numerous problems, including that it says that there is no disc inserted (when there is), that the files are corrupted, that there are no setup files on the disk, or that these setup files can't be used.

I'm not sure why - I know there is nothing wrong with the disk, as it can install on another computer.
Any advice on what to do?

Try using the drive with another disc you know to be working. Your disc drive may need to be cleaned or replaced.